Period Integrals of Hypersurfaces via Tropical Geometry

Abstract

Let $\left \{ Z_{t} \right \}_{t}$ be a one-parameter family of complex hypersurfaces of dimension $d \geq 1$ in a toric variety. We compute asymptotics of period integrals for $\left \{ Z_{t} \right \}_{t}$ by applying the method of Abouzaid-Ganatra-Iritani-Sheridan, which uses tropical geometry. As integrands, we consider Poincar & eacute; residues of meromorphic $(d+1)$ -forms on the ambient toric variety, which have poles along the hypersurface $Z_{t}$ . The cycles over which we integrate them are spheres and tori, which correspond to tropical $(0, d)$ -cycles and $(d, 0)$ -cycles on the tropicalization of $\left \{ Z_{t} \right \}_{t}$ , respectively. In the case of $d=1$ , we explicitly write down the polarized logarithmic Hodge structure of Kato-Usui at the limit as a corollary. Throughout this article, we impose the assumption that the tropicalization is dual to a unimodular triangulation of the Newton polytope.